Pythagoras' Theorem
Pythagoras' Theorem is one of the most celebrated results in all of mathematics, stating a precise relationship between the three sides of any right-angled triangle. Specifically, if a triangle contains a 90° angle, then the square of the length of the hypotenuse — the side directly opposite the right angle — equals the sum of the squares of the other two sides. This single equation unlocks a vast array of geometric and practical problems, from calculating distances on a map to designing buildings and analysing vectors. Although the theorem bears the name of the ancient Greek mathematician Pythagoras of Samos (c. 570–495 BC), evidence of the result appears in Babylonian clay tablets dating back to around 1800 BC, and Egyptian rope-stretchers used 3-4-5 triangles to construct right angles in the field. What Pythagoras and his school are credited with is providing the first rigorous deductive proof, placing the result within a logical framework rather than treating it as a practical rule of thumb. This marks a foundational moment in the history of deductive reasoning. The theorem sits at the heart of Euclidean geometry and has hundreds of known proofs, ranging from Euclid's own area-based argument in the Elements to elegant algebraic proofs using similar triangles, and even a proof famously sketched by US President James Garfield in 1876. Its converse is equally important: if the square of the longest side of a triangle equals the sum of the squares of the other two sides, the triangle must contain a right angle. Mastering this theorem and its applications is essential for success in geometry, trigonometry, coordinate geometry, and beyond.
- Identifying the Hypotenuse
- Stating the Theorem
- A Proof by Similar Triangles
- Finding a Missing Side
- The Converse of Pythagoras' Theorem
Identifying the Hypotenuse
In any right-angled triangle, label the right angle with a small square symbol. The side opposite this right angle is the longest side and is called the hypotenuse. The other two sides are called the legs (or catheti). It is critical to identify the hypotenuse correctly before applying the theorem, because it always appears alone on one side of the equation.
Stating the Theorem
If the legs have lengths a and b, and the hypotenuse has length c, then a² + b² = c². This means: build a square on each side of the triangle; the area of the square on the hypotenuse exactly equals the combined areas of the squares on the two legs. This area interpretation is exactly how Euclid proved the theorem in Book I, Proposition 47 of the Elements.
A Proof by Similar Triangles
Drop an altitude from the right-angle vertex perpendicular to the hypotenuse, splitting the triangle into two smaller triangles. Each smaller triangle is similar to the original (all three angles match). From the similarity ratios: a/c = p/a gives a² = cp, and b/c = q/b gives b² = cq, where p and q are the two segments of the hypotenuse (so p + q = c). Adding: a² + b² = cp + cq = c(p+q) = c². This is a clean algebraic proof from first principles.
Finding a Missing Side
To find the hypotenuse: c = √(a² + b²). To find a missing leg: a = √(c² − b²). Always check that c is the largest value — if you compute a leg and get a value larger than the hypotenuse, you have made an error. Also verify that c² > a² and c² > b² individually, confirming your answer is geometrically sensible.
The Converse of Pythagoras' Theorem
The converse states: if a² + b² = c² for the sides of a triangle (with c the longest), then the triangle is right-angled. This allows us to test whether a triangle is right-angled without measuring any angles. For example, a triangle with sides 5, 12, 13 satisfies 5² + 12² = 25 + 144 = 169 = 13², so it is a right-angled triangle. Common Pythagorean triples (integer solutions) include (3,4,5), (5,12,13), (8,15,17), and (7,24,25).
The Key Equation
- a — First leg (length (e.g. cm, m)): The length of one of the two shorter sides of the right-angled triangle.
- b — Second leg (length (e.g. cm, m)): The length of the other shorter side of the right-angled triangle.
- c — Hypotenuse (length (e.g. cm, m)): The length of the longest side, always opposite the right angle. This is the side isolated in the equation.
Worked Example
Problem. A right-angled triangle has legs of length 9 cm and 40 cm. Find the exact length of the hypotenuse. Then verify the answer is reasonable by checking it is greater than both legs.
Answer. The hypotenuse is 41 cm.
Origins · 1 minPythagoras of Samos and how this was discovered
Pythagoras founded a philosophical and mathematical school in Croton (southern Italy) whose members, the Pythagoreans, are credited with the first rigorous deductive proof of the right-triangle relationship now bearing his name. The school treated number and geometry as the foundation of all reality, profoundly influencing Plato and the entire Western mathematical tradition. Pythagoras is also credited with early work on musical ratios, the theory of means, and the study of odd and even numbers. The Pythagoreans reportedly kept their discovery of irrational numbers — numbers like √2 that cannot be expressed as a fraction — a closely guarded secret, as it threatened their philosophical belief that all quantities could be expressed as ratios of whole numbers. According to legend, a member named Hippasus was drowned for revealing the secret to outsiders.
Related Concepts
- Trigonometric Ratios in Right Triangles — Sine, cosine, and tangent ratios are defined using the sides of a right-angled triangle. Pythagoras' Theorem underpins the fundamental identity sin²θ + cos²θ = 1, connecting trigonometry directly to this theorem.
- The Distance Formula in Coordinate Geometry — The distance between two points (x₁,y₁) and (x₂,y₂) is derived directly from Pythagoras' Theorem applied to a right triangle formed by the horizontal and vertical separations of the points.
- Similar Triangles — One of the cleanest proofs of Pythagoras' Theorem uses the properties of similar triangles formed by dropping an altitude to the hypotenuse, making similarity an essential supporting concept.
- Angle Properties of Triangles — Understanding that angles in a triangle sum to 180° and knowing how to identify a right angle is prerequisite knowledge for correctly applying Pythagoras' Theorem.